\newproblem{lay:5_2_23}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 5.2.23}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Show that if $A=QR$ with $Q$ invertible, then $A$ is similar to $A_1=RQ$.
}{
   % Solution
	We remind that the matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ such that $B=P^{-1}AP$ (with $A,B,P \in \mathcal{M}_{n\times n}$).
	This means that we need to find an invertible matrix $P$ such that
	\begin{center}
		$A_1=P^{-1}AP$ \\
		$RQ=P^{-1}QRP$
	\end{center}
	If we make $P=Q$, since $Q$ is invertible, we have $P^{-1}=Q^{-1}$ and
	\begin{center}
		$RQ=Q^{-1}QRQ=RQ$
	\end{center}
	So, we have proven that $A$ and $A_1$ are similar.
}
\useproblem{lay:5_2_23}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
